Hypergraphs with Polynomial Representation: Introducing $r$-splits

TL;DR

This paper introduces the concept of r-splits in graphs, demonstrating that their associated hypergraphs can be efficiently represented with polynomially many hyperedges despite potential exponential complexity.

Contribution

It formalizes r-splits as hypergraphs and proves they can be represented with O(n^{r+1}) hyperedges, providing bounds and a generalization of set orthogonality.

Findings

Hypergraphs of r-splits can be represented with O(n^{r+1}) hyperedges.

Existence of hypergraphs requiring at least Ω(n^r) hyperedges for representation.

Introduction of r-splits as a new concept in graph decomposition.

Abstract

Inspired by the split decomposition of graphs and rank-width, we introduce the notion of $r$-splits. We focus on the family of $r$-splits of a graph of order $n$, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only $\mathcal O(n^{r+1})$ of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least $\Omega(n^r)$ hyperedges to be represented, using a generalization of set orthogonality.